We are given two parallel lines $3x + 4y + c_1 = 0$ and $3x + 4y + c_2 = 0$, with $c_2 > c_1 > 0$, and we are also given the following information:

1. The distance between the two parallel lines is 4.
2. The minimum distance between the point $(2,3)$ and the line $3x + 4y + c_1 = 0$ is 6.

We are tasked with finding the value of $c_1 + c_2$.

Step 1: Formula for distance between two parallel lines

The general formula for the distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is:

$\text{Distance} = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}$

For the lines given, $A = 3$, $B = 4$, and the distance between the two lines is 4. Therefore, we have:

$4 = \frac{|c_2 - c_1|}{\sqrt{3^2 + 4^2}} = \frac{|c_2 - c_1|}{5}$

Multiplying both sides by 5, we get:

$|c_2 - c_1| = 20$

Since $c_2 > c_1$, we conclude that:

$c_2 - c_1 = 20$

Step 2: Formula for distance between a point and a line

The formula for the distance between a point $(x_1, y_1)$ and a line $Ax + By + C = 0$ is:

$\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

We are given that the distance between the point $(2, 3)$ and the line $3x + 4y + c_1 = 0$ is 6. Using this formula, we substitute $A = 3$, $B = 4$, $x_1 = 2$, $y_1 = 3$, and $C = c_1$:

$6 = \frac{|3(2) + 4(3) + c_1|}{\sqrt{3^2 + 4^2}} = \frac{|6 + 12 + c_1|}{5}$

Simplifying the equation:

$6 = \frac{|18 + c_1|}{5}$

Multiplying both sides by 5:

$30 = |18 + c_1|$

This gives us two possible cases:

1. $18 + c_1 = 30$, which gives $c_1 = 12$.
2. $18 + c_1 = -30$, which gives $c_1 = -48$.

Since we are told that $c_1 > 0$, we conclude that:

$c_1 = 12$

Step 3: Finding $c_2$

We already know that $c_2 - c_1 = 20$, so:

$c_2 = c_1 + 20 = 12 + 20 = 32$

Step 4: Finding $c_1 + c_2$

Finally, we can calculate:

$c_1 + c_2 = 12 + 32 = 44$

Thus, the value of $c_1 + c_2$ is $\boxed{44}$.

---

Would you like any further details or explanations?

Here are five related questions to expand your understanding:

1. How do you derive the distance formula between two parallel lines?
2. Can the formula for distance from a point to a line be applied to non-linear curves?
3. What if $c_2 < c_1$? How would the result change?
4. How does the distance between parallel lines relate to their slopes?
5. Can this method be extended to 3D space, and how would the formulas change?

Tip: For solving geometry problems involving distances, always remember to apply the correct distance formula depending on whether it’s between points, lines, or a combination of both.